>Let $A, B, C$ be three points in xy-plane, whose position vector are given by $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $a \hat{i}+(1-a) \hat{j}$ respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between the vectors $\overrightarrow{\mathrm{OA}}$ and $\overrightarrow{\mathrm{OB}}$ is $\frac{9}{\sqrt{2}}$, then the sum of all the possible values of $a$ is :
Previous 10 Questions — JEE Main 2025 (28 January Evening Shift)
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If $\alpha + i\beta$ and $\gamma + i\delta$ are the roots of $x^2 - (3 - 2i)x - (2i - 2) = 0$, $i = \sqrt{-1}$, then $\…
Topic: JEE Main 2025 (28 January Evening Shift)
If $\sum\limits_{r=1}^{13}\left\{\frac{1}{\sin \left(\frac{\pi}{4}+(r-1) \frac{\pi}{6}\right) \sin \left(\frac{\pi}{4}+…
Topic: JEE Main 2025 (28 January Evening Shift)
If A and B are the points of intersection of the circle $x^2 + y^2 - 8x = 0$ and the hyperbola $\frac{x^2}{9} - \frac{y…
Topic: JEE Main 2025 (28 January Evening Shift)
Let S be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set S, o…
Topic: JEE Main 2025 (28 January Evening Shift)
Let [x] denote the greatest integer less than or equal to x. Then the domain of $ f(x) = \sec^{-1}(2[x] + 1) $ is:
Topic: JEE Main 2025 (28 January Evening Shift)
Let $f:\mathbb{R}-{0}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\l…
Topic: JEE Main 2025 (28 January Evening Shift)
If the components of $\vec a=\alpha,\hat i+\beta,\hat j+\gamma,\hat k$ along and perpendicular to $\vec b=3\hat i+\hat …
Topic: JEE Main 2025 (28 January Evening Shift)
Let the coefficients of three consecutive terms $T_r, T_{r+1}$ and $T_{r+2}$ in the binomial expansion of $(a+b)^{12}$ …
Topic: JEE Main 2025 (28 January Evening Shift)
Two equal sides of an isosceles triangle are along $-x+2y=4$ and $x+y=4$. If $m$ is the slope of its third side, then t…
Topic: JEE Main 2025 (28 January Evening Shift)
The square of the distance of the point $\left(\dfrac{15}{7},,\dfrac{32}{7},,7\right)$ from the line $\dfrac{x+1}{3}=\d…
Topic: JEE Main 2025 (28 January Evening Shift)
Next 10 Questions — JEE Main 2025 (28 January Evening Shift)
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If $f(x)=\displaystyle\int \frac{1}{x^{1/4}\left(1+x^{1/4}\right)},dx,; f(0)=-6$, then $f(1)$ is equal to:
Topic: JEE Main 2025 (28 January Evening Shift)
Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin…
Topic: JEE Main 2025 (28 January Evening Shift)
Let $f$ be a real valued continuous function defined on the positive real axis such that $g(x)=\int\limits_0^x t f(t) d…
Topic: JEE Main 2025 (28 January Evening Shift)
et $f:[0,3]\to A$ be defined by
$,f(x)=2x^3-15x^2+36x+7,$
and $g:[0,\infty)\to B$ be defined by
$,g(x)=\dfrac{x^{2025}}…
Topic: JEE Main 2025 (28 January Evening Shift)
For positive integers $n$, if
$4a_n=(n^2+5n+6)$
and
$S_n=\displaystyle\sum_{k=1}^{n}\frac{1}{a_k},$
then the value of $…
Topic: JEE Main 2025 (28 January Evening Shift)
